The paper introduces an adaptive version of the stabilized Trace Finite Element Method (TraceFEM) designed to solve low-regularity elliptic problems on level-set surfaces using a shape-regular bulk mesh in the embedding space. Two stabilization variants, gradient-jump face and normal-gradient volume, are considered for continuous trace spaces of the first and second degrees, based on the polynomial families and . We propose a practical error indicator that estimates the ‘jumps’ of finite element solution derivatives across background mesh faces and it avoids integration of any quantities along implicitly defined curvilinear edges of the discrete surface elements. For the family of piecewise trilinear polynomials on bulk cells, the solve-estimate-mark-refine strategy, combined with the suggested error indicator, achieves optimal convergence rates typical of two-dimensional problems. We also provide a posteriori error estimates, establishing the reliability of the error indicator for the and elements and for two types of stabilization. In numerical experiments, we assess the reliability and efficiency of the error indicator. While both stabilizations are found to deliver comparable performance, the lowest degree finite element space appears to be the more robust choice for the adaptive TraceFEM framework.

中文翻译：

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表面偏微分方程的自适应稳定迹有限元方法


本文介绍了稳定跟踪有限元方法 (TraceFEM) 的自适应版本，旨在使用嵌入空间中的形状规则体网格来解决水平集曲面上的低规则性椭圆问题。基于多项式族 和 ，对于一次和二次的连续迹空间，考虑了两种稳定变体，梯度跳跃面和法向梯度体积。我们提出了一种实用的误差指示器，用于估计跨背景网格面的有限元解导数的“跳跃”，并且它避免了沿着离散表面元素的隐式定义的曲线边缘的任何量的积分。对于体单元上的分段三线性多项式族，求解-估计-标记-细化策略与建议的误差指标相结合，实现了二维问题典型的最佳收敛速度。我们还提供后验误差估计，建立 和 元素以及两种类型的稳定性的误差指标的可靠性。在数值实验中，我们评估误差指示器的可靠性和效率。虽然两种稳定方法都可以提供相当的性能，但最低阶有限元空间似乎是自适应 TraceFEM 框架更稳健的选择。